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Symmetry in Lie Groups and Lie Algebras

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Special Issue Editors

Prof. Dr. Liangyun Chen

E-Mail Website
Guest Editor

School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China
Interests: Lie algebra; Lie superalgebra; hom-type algebra

Dr. Yao Ma

E-Mail Website
Guest Editor

School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China
Interests: Lie algebra; Lie superalgebra; quantum group

Special Issue Information

Dear Colleagues,

Symmetry is a fundamental concept in mathematics and physics and is deeply connected to the study of Lie groups and Lie algebras. Lie groups, which are smooth manifolds with group structures, and their associated Lie algebras, which describe infinitesimal symmetries, provide powerful tools for understanding continuous symmetries in geometry, differential equations, quantum mechanics, and theoretical physics.

This Special Issue aims to explore the interplay between symmetry principles and Lie theoretic methods, highlighting their applications in pure and applied mathematics, mathematical physics, and related disciplines. Topics may include theoretical developments, computational approaches, and applications in fields such as integrable systems, gauge theories, and geometric mechanics.

We look forward to your submissions.

Prof. Dr. Liangyun Chen
Dr. Yao Ma
Guest Editors

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Published Papers (3 papers)

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Research

17 pages, 270 KB

Open AccessArticle

Symmetric Structures in Mock-Lie Algebras: The Quasi-Centroid and Its Matrix Representations up to Dimension 5

by Keli Zheng, Yue Zhu, Wei Shan and Ying Guo

Symmetry 2025, 17(12), 2080; https://doi.org/10.3390/sym17122080 - 4 Dec 2025

Viewed by 135

Abstract

Symmetric structures are key in non-associative algebras. A Mock-Lie algebra, defined by commutativity and the Jacobi identity, shows strong algebraic symmetry. This paper studies the quasi-centroid, which captures the symmetry of linear operators commuting with the algebra’s product. We define the quasi-centroid and set its condition for linear endomorphisms under the bracket operation. We classify matrix representations of quasi-centroids for all Mock-Lie algebras of dimensions 2 to 5 by computing matrices and analyzing coefficient relations. These results provide a foundation for further structural study. We also show that in each case, the centroid is strictly contained in the quasi-centroid, confirming proper containment for all these algebras. Full article

(This article belongs to the Special Issue Symmetry in Lie Groups and Lie Algebras)

16 pages, 292 KB

Open AccessArticle

On the Classification of Totally Geodesic and Parallel Hypersurfaces of the Lie Group Nil⁴

by Guixian Huang and Jinguo Jiang

Symmetry 2025, 17(11), 1979; https://doi.org/10.3390/sym17111979 - 16 Nov 2025

Viewed by 194

Abstract

This work establishes a complete algebraic classification of hypersurfaces with totally symmetric cubic form, including the Codazzi, parallel, and totally geodesic cases, on the 4-dimensional 3-step nilpotent Lie group

N i l^{4}

endowed with six left-invariant Lorentzian metrics. Combined with prior results, [...] Read more.

N i l^{4}

endowed with six left-invariant Lorentzian metrics. Combined with prior results, we achieve a complete classification of such hypersurfaces on 4-dimensional nilpotent Lie groups. The core of our approach lies in the explicit derivation and solution of the Codazzi tensor equations, which directly leads to the construction of these hypersurfaces and provides their explicit parametrizations. Our main results establish the existence of Codazzi hypersurfaces on

N i l^{4}

, demonstrate the non-existence of totally geodesic hypersurfaces, specify the algebraic condition for a Codazzi hypersurface to become parallel, and provide their explicit parametrizations. This observation highlights fundamental differences between Lorentzian and Riemannian settings within hypersurface theory. This work thus clarifies the distinct geometric properties inherent to the Lorentzian cases on nilpotent Lie groups. Full article

(This article belongs to the Special Issue Symmetry in Lie Groups and Lie Algebras)

24 pages, 392 KB

Open AccessArticle

Supercommuting Maps on Incidence Algebras with Superalgebra Structures

by Nof T. Alharbi

Symmetry 2025, 17(11), 1817; https://doi.org/10.3390/sym17111817 - 28 Oct 2025

Viewed by 367

Abstract

Let R be a 2-torsion-free and

n!

-torsion-free commutative ring with unity, and let X be a locally finite preordered set. We endow the incidence algebra

I (X, R)

with a superalgebra structure via a nontrivial idempotent, which decomposes [...] Read more.

Let R be a 2-torsion-free and

n!

-torsion-free commutative ring with unity, and let X be a locally finite preordered set. We endow the incidence algebra

I (X, R)

with a superalgebra structure via a nontrivial idempotent, which decomposes

I (X, R)

into even and odd parts

A_{0} \oplus A_{1}

. Our main result shows that if any two directed edges in each connected component of the complete Hasse diagram

(X, D)

lie in one cycle, then every supercommuting map on

I (X, R)

is proper. A supercommuting map

θ : I (X, R) \to I (X, R)

is defined by the condition

{[θ (x), x]}_{s} = 0

for all

x \in I (X, R)

, where

{[a, b]}_{s} = a b - {(- 1)}^{| a | | b |} b a

is the supercommutator. We prove that such maps must take the form

θ (x) = λ x + μ (x),

where

λ \in Z_{s} (I (X, R))

(the supercenter) and

μ : I (X, R) \to Z_{s} (I (X, R))

is an R-linear map. This generalizes the known results on commuting maps of incidence algebras and other associative algebras. Full article

(This article belongs to the Special Issue Symmetry in Lie Groups and Lie Algebras)

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